President Bush, following up on a stern public message from Secretary of State James A. Baker III last week, has written a "lengthy" letter to Israeli Prime Minister Yitzhak Shamir questioning the willingness of his new conservative government to revive the Middle East peace process.

White House spokesman Marlin Fitzwater said the letter was sent in the last few days and was both a congratulatory letter on the formation of the government and a lengthier discussion of U.S.-Israeli relations.

Administration officials said Bush reiterated U.S. commitment to a plan that Shamir espoused and the administration adopted last year for the election of Palestinian representatives in the occupied territories. But Shamir vehemently opposes a formula proposed by Baker for opening Israeli-Palestinian talks.

U.S. frustration with months of effort to get the peace talks moving erupted publicly last week when Baker, appearing before a congressional committee, outlined steps the administration had taken and noted statements from the new government signaling opposition to the peace plan.

Giving the White House phone number, Baker said, "When you are serious about peace, call us."

Officials said Bush's letter posed questions to Shamir about his government's positions on various aspects of the peace talks and expressed U.S. interest in continuing to work for progress.

Israeli news reports quoted Shamir as saying yesterday in a television appearance that Bush had asked "for replies and explanations about our path in the framework of the peace process." Shamir was quoted as saying that if there are differences, "I sincerely hope . . . we will reach agreement."

In his 1989 initiative, Shamir proposed elections among Palestinian Arabs in the West Bank and Gaza to choose those who would negotiate with Israel the conditions of a five-year limited self-rule period. A second stage of talks would address an overall Israeli-Arab settlement.

Baker devised a formula to implement the Shamir plan and it was this formula to which Shamir objected. The Israeli government collapsed in March and it was not until last week that a new government was formed, one considered the most conservative in that nation's history.

In an interview with Knight-Ridder newspapers, Bush said he had given Shamir some "suggestions" he said he hoped would clear the way for resumption of peace talks, but he did not elaborate. "We aren't going to pick up our marbles and go home," Bush said.